1. M. Zareinejad

4.  Use haptic device to physically interact with the VE – optical encoders measure position of end effector – actuators apply forces to the user – haptic rendering algorithms compute such forces given the new positions

7.  “Haptic rendering is the process of computing and generating forces in response to user interaction with virtual environment”  Computing forces and torques that should be applied to the tip of the haptic display in order to represent forces of a physical phenomena or represent some data

13.  Virtual Wall “algorithm”

19.  In vector calculus, a vector field is an assignment of a vector to each point in space

27.  ‘God-object’ is an ideal interaction point.  ƒ It stays on the surface when the object penetration occurs.  ƒ It locates on the nearest surface from the HIP.

36.  An active surface is the surface that has the God-object on it.  ƒ To be an active surface, The God-object must be located in positive distance from the surface. The HIP must be located in negative distance from the surface.

37.  Draw a line from the old God-object to the new HIP.  If the line is not under three edges, the surface is not active now.  Change the active surface after a cycle.

38.  To be an active surface, the HIP must located in negative distance from the surface.  ƒ On acute concave object, the God-object moves below the surface but the HIP does not.

39.  I. Find a new God-object location.  II. Using this as a HIP, check whether there is a new constraint.  III. If there is a new constraint, find “new” God-object location.  IV. Continue until no new constraint is found.

40.  We find point Q with the minimal distance.  This is a new God-object location.

41.  For three active plane constraints,  This can be solved using the Lagrange Multiplier Theorem.

42.  Set Lagrangian as By the Lagrange Multiplier Theorem

43.  We can compute x, y and z in at most 65 of ×and ÷operations.  ƒ Lower number of constraints make computation much faster.